Shape calculus and finite element method in smooth domains

Mathematics of Computation

Volumn 70, Issue 233, 2001, Pages 1-15

  Tiihonen, T ^a  

^a UNIVERSITY OF JYVÄSKYLÄ   (Finland)

Author keywords

Continuous dependence on geometry; Curved boundary; Error estimates; Finite elements; Shape derivatives

Indexed keywords

EID: 0035624933     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0025-5718-00-01323-5     Document Type: Article

References (17)

1. Babuska I., Stabilität des Definitionsgebietes mit Rücksicht auf grundlegende Probleme der Theorie des partiellen Differentialgleichungen auch in Zusammenhang mit der Elasticitätstheorie, Czechoslovak Math. J. 11, 1961, pp. 76-105, 165-203. MR 23:A2629
2. Babuska I., The theory of small changes in the domain of existence in the theory of partial differential equations and its applications, in Differential equations and their applications, Academic Press, New York. 1963, pp. 13-26. MR 53:7070
3. Berger A., Scott R., Strang G., Approximate boundary conditions in the finite element method, in Symposia Mathematica, Vol X, Academic Press, London, 1972, pp. 295-313. MR 53:7070
4. Ciarlet P.G., Raviart P.A., Interpolation theory over curved elements with applications to finite element methods, Comp. Meth. Appl. Mech. Eng., 1, 1972, pp. 217-249. MR 51:11991
5. Lenoir M., Optimal isoparametric finite elements and error estimates for domains involving curved boundaries, SIAM J. Num. Anal., 1986, pp. 562-580. MR 87m:65163
6. Rannacher R., Scott R., Some optimal error estimates for piecewise linear finite element approximations, Math. Comp. 38, 1982, pp. 437-445. MR 83e:65180
7. Scott R. Interpolated boundary conditions in the finite element method, SIAM J. Num. Anal., 12, 1975, pp. 404-427. MR 52:7162
8. 0-projection into finite element spaces, Math. Comp. 38, 1982, pp. 1-22. MR 82m:65106
9. Sokolowski J., Zolésio J.P., Introduction to shape optimization, shape sensitivity analysis, Springer, Berlin, 1992. MR 94d:49002
10. Strang G., Berger A. E., The change in solution due to change in domain, Proceedings of symposia in pure mathematics, XXIII, Ed. D.C. Spencer, AMS, vol. 23, 1973. pp. 199-205. MR 49:1796
11. Thomée V., Polygonal domain approximation in Dirichlet's problem, Journal of IMA, 11, 1973, pp. 33-44. MR 50:1538
12. Tiihonen T., Finite element approximation of nonlocal heat radiation problems, Math. Mod. Meth. Appl. Sci., 8, 1998, pp. 1071-1089. MR 99h:65172
13. Tiihonen T., Shape calculus and FEM in smooth domains, in Finite Element Methods: Superconvergence, Post-processing and A Posteriori Estimates, eds. Křížek M., Neittanmäki P., Stenberg, R., Marcel Dekker, New York, 1997, pp. 259-267. MR 99c:49041
14. Verfürth R., Mixed finite element approximation of a fluid flow problem, in The mathematics of finite elements and applications V, ed. J. Whiteman, Academic Press, 1985, pp. 335-342. MR 87a:65184
15. Verfürth R., Finite element approximation of incompressible Navier-Stokes equations with slip boundary condition. II, Num. Math. 59, 1991, pp. 616-636. MR 92k:65170
16. Zlamal M., Curved elements in the finite element method, I, II, SIAM J. Num. Anal., 10, 1973, pp. 229-240, 11, 1974, pp. 347-362. MR 52:16060; MR 49:8400
17. Zlamal M., Curved elements in the finite element method, I, II, SIAM J. Num. Anal., 10, 1973, pp. 229-240, 11, 1974, pp. 347-362. MR 52:16060; MR 49:8400